When Assumption Smooth GAN holds, the
mean value theorem implies that the Lipschitz condition \(|F(x,z,\theta _{1})-F(x,z,\theta _{2})|\le \sup _{\theta \in \Theta }|\partial F(x,z,\theta )/\partial \theta |\,|\theta _{1}-\theta _{2}|\) 
holds for all \(\theta _{1},\theta _{2}\in \Theta \)  where \(\mathbb {E}[\sup _{\theta \in \Theta }|\partial F(X,Z,\theta )/\partial \theta |^{2}]<\infty \) 
by assumption. Example 19.7 and Theorem 19.5 of [1]}
now imply the weak convergence in Assumption 4;
moreover, it holds that \(\mathbb {E}[\sup _{\theta \in \Theta }(F(X,Z,\theta ))^{2}]<\infty \) 
([2]}).
Condition \(\inf _{\theta \in \Theta _{0}}\mathbb {E}[(\mathbb {G}(\theta ))^{2}]>0\) 
follows from the requirement \(\inf _{\theta \in \Theta }\operatorname{Var}[F(X,Z,\theta )]>0\) 
in Assumption Smooth GAN.
Therefore Assumptions 1(b) and 4 hold.
